H∞ state estimation of continuous-time neural networks with uncertainties

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\infty }$$\end{document}H∞ state estimation is addressed for continuous-time neural networks in the paper. The norm-bounded uncertainties are considered in communication neural networks. For the considered neural networks with uncertainties, a reduced-order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\infty }$$\end{document}H∞ state estimator is designed, which makes that the error dynamics is exponentially stable and has weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\infty }$$\end{document}H∞ performance index by Lyapunov function method. Moreover, it is also given the devised method of the reduced-order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\infty }$$\end{document}H∞ state estimator. Then, considering that sampling the output y(t) of the neural network at every moment will result in waste of excess resources, the event-triggered sampling strategy is used to solve the oversampling problem. In addition, a devised method is also given for the event-triggered reduced-order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\infty }$$\end{document}H∞ state estimator. Finally, by the well-known Tunnel Diode Circuit example, it shows that a lower order state estimator can be designed under the premise of maintaining the same weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\infty }$$\end{document}H∞ performance index, and using the event-triggered sampling method can reduce the computational and time costs and save communication resources.


Problem statements and preliminaries
Consider continuous-time neural networks with uncertainties as following: where e(t) ∈ R n is the system state, f (e(t)) = f 1 (e 1 (t)) f 2 (e 2 (t)) • • • f n (e n (t)) T ∈ R n is the neuron activation function, y(t) ∈ R m is the measured output, z(t) ∈ R p is the estimated signal, v(t) ∈ R q denotes the Gaussian white noise, v(t) ∈ L 2 [0, ∞) .A ∈ R n×n , B ∈ R n×n , C ∈ R n×q , D ∈ R m×n , F ∈ R m×n , and G ∈ R m×q are the known constant matrices.�A(t) , �B(t) represent the system time-varying uncertainties which are unknown matrices and subject to the below constraints where M ∈ R n×n , L 1 , L 2 ∈ R n×n are matrices of set constants, and N(t) satisfying N T (t)N(t) ≤ I is an unknown time-varying matrix.For ease of notation, set �A(t) �A and �B(t) �B.Then for the system (1), the H ∞ state estimator is formed as in which e f (t) ∈ R n f is the state of estimator and , and D f ∈ R p×m are the unknown filter matrices.And 1 ≤ n f ≤ n , when n f = n , Eq. ( 3) is called the full-order H ∞ state estimator of Eq. ( 1); when 1 ≤ n f < n , Eq. ( 3) is called the reduced-order H ∞ state esti- mator of Eq. (1).And the structure of the plant is shown in Fig. 1.From Eqs. ( 1) and (3), it is gotten the error dynamics where ξ(t) = e(t) e f (t) And the lemmas and definition are also proposed as following.
Definition 1 For a preassigned constant γ > 0 , the system (4) is called to be exponentially stable and have a weighted H ∞ performance index γ , if it satiafies (1) when v(t) = 0 , the system (4) is exponentially stable; (2) under zero initial condition (ZIC), one holds

Main results
The H ∞ state estimation issue here is to devise the state estimator matrices A f , B f , C f , and D f to make sure the system (4) be exponentially stable and have a weighted H ∞ performance index.The main results are as following.
Theorem 1 For the system (4) and the given positive constant γ 0 , suppose there exist positive scalars ι 1 , ι 2 , and a positive symmetric matrices Proof Let the Lyapunov function be then it obtains in which a = min{ min (P 1 ), min (P 2 )} , b = max{ max (P 1 ), max (P 2 )}.
On the other hand, consider that sampling y at all times exists to sample unnecessary data.To determine the specific values of the outputs y(t) and reduce the number of samples, an event-triggered sampling strategy is used to generate the sampling time series {t k } , t 0 = 0: where e y (t) = y(t) − y(t k ) , t > t k , � > 0 and � > 0 are event-triggered parameters to be designed.And in the event-triggered sampling interval [t k , t k+1 ) , set ŷ(t) = y(t k ) , where ŷ(t) is the output of Zero Order Holder (ZOH) in Fig. 2. Then for the system (1), the form of the H ∞ state estimator based on the event-triggered sam- pling strategy (12) is where e f (t) , z f (t) , A f , B f , C f , and D f are same as them in (3).
From Eqs. ( 1), ( 12) and ( 13), it is gotten the error dynamics where ξ(t), Ã, Ã, B, B, C, H, and ẽ(t) are same as them in (4), and Then, based on the event-triggered sampling strategy (Eq.12), the solution of H ∞ state estimation issue is as following: Theorem 2 For the system (14) and the given positive constant γ 0 , suppose there exist positive scalars ι 1 , ι 2 , and a positive symmetric matrices www.nature.com/scientificreports/hold, then the system (14) be exponentially stable and have a weighted H ∞ performance index γ 0 .And the state estimator matrices are Proof Also select the Lyapunov function as Eq. ( 6).Only the proof of the following equation is given here, and the rest of the proof procedure is similar to the proof of Theorem 1.
By the event-triggered sampling strategy (Eq.12), when t ∈ [t k , t k+1 ) , the event is not triggered, it means that i.e.
In [t k , t k+1 ) , according to Eqs. ( 13), ( 14) and (17), and taking the time derivation of Lyapunov function ( 6) along ( 14) could be obtained that w h e r e f (e(t)) The last step in Eq. ( 18) also utilizes Lemma 2, which we would not repeat here.Then, using the Schur Complement Lemma three times for Eq. ( 15) and collapsing yields �1 + �T 2 �2 + �T 3 � �3 < 0 , and thus Eq. ( 16) holds.The proof is completed.Theorem 3 For the system (14) if it can find positive scalars ι 1 , ι 2 , and a positive symmetric matrices P 1i ∈ R n×n , and D F ∈ R p×m such that then it could be find a suboptimal H ∞ performance index γ 0 .
In addition, to avoid triggering the sample an infinite number of times in a short period of time, i.e.Zeno behavior, the following theorem will give a positive lower bound on the event trigger interval.

r, then the lower bounded of the minimum event-trigger inter-execution interval is
where Proof For any 0 < t ∈ [t k , t k+1 , then �e y (t)� = �y(t) − y(t k )� < ����y(t k )� from (12).For any t ∈ [t k , t k+1 ) , it holds Taking the time integral from t k to t, one has When t = t k+1 , it triggers the homologous event, that signifies And by �e y (t k )� = 0 and Eq. ( 19), it follows Suppose �x(t k )� � = 0 , then it yields that In summary, one can find a T min = min{t k+1 − t k } > 0 that excludes Zeno behavior under the proposed event- triggered strategy (12).Proof is finished.
Remark 1 Theorem 3 serves to rule out the possibility that the event-triggered strategy (Eq.12) may have unlimited sampling for a short period of time, i.e., the Zeno phenomenon.If infinite sampling occurs in a short period of time, this will not only not reduce the number of samples, but also increase the burden of computer computation and measurement, or even computer crash, so the event-triggered strategy that excludes the occurrence of such a situation is good for in application of the considered event-triggered strategy.

Simulation
The well-known Tunnel Diode Circuit in Fig. 3 which modeled as the system (1) is illustrated to display the effectiveness of the method.
Assume the coefficient matrices are min γ 0 s.t.(15) holds, Choose the neuron activation function as f (e(t)) = tanh(e 1 ) tanh(e 2 ) , and By Theorem 1, it can formulate the state estimator parameters as following: (1) The parameters of the full-order ( n f = 2 ) H ∞ state estimator for the system (1): (2) The parameters of the reduced-order ( n f = 1 ) H ∞ state estimator for the system (1): And by Theorem 2, it can formulate the state estimator parameters as following: (1) The parameters of the full-order ( n f = 2 ) H ∞ state estimator for the system (1): 2059, and the event- triggered parameters are = 1.9985 and = 0.0343.(2) The parameters of the reduced-order ( n f = 1 ) H ∞ state estimator for the system (1): 4845, and the event-triggered parameters are = 0.6467 and = 0.2760.
Then, by the SIMULINK Toolbox of MATLAB, the system (4) is exponentially stable with v(t) = 0 , see Fig. 4a, b by Theorem 1, Fig. 5a, b by Theorem 2. And from Fig. 4a, 5a for n f = 2 , Figs. 4b, 5b for n f = 1 , the locus of ξ(t) of system (4) with v(t) = 0 shows little difference in the overall trend.The oscilloscopes for z(t) and z f (t) is displayed in Fig. 4c, d  From Figs. 4c, d, 5c, d above, compared with the full-order ( n f = 2 ) and reduce-order ( n f = 1 ) state estimators, the z f (t) of the reduce-order state estimator is closer to z(t), and the order is lower.This shows that a lower order state estimator can be designed under the premise of maintaining the same weighted H ∞ performance index.And according to Figs. 4c, 5c, 4d and 5d, the state estimators given by Theorems 1 and 2 do not differ much in their estimation of z(t).
From Figs. 4c, d, 5c, d above, compared with the full-order ( n f = 2 ) and reduce-order ( n f = 1 ) state estimators, the z f (t) of the reduce-order state estimator is closer to z(t), and the order is lower.This shows that a lower order state estimator can be designed under the premise of maintaining the same weighted H ∞ performance index.And according to Figs. 4c, 5c, 4d and 5d, the state estimators given by Theorems 1 and 2 do not differ much in their estimation of z(t).
However, as can be seen from Fig. 6a, b, the use of event-triggered sampling avoids the need to sample y(t) from time to time, which reduces the computational and time costs and saves resources.And compared with the existing studies, such as 1,2,4,5 , the reduce-order ( n f = 1 ) state estimator and the event-triggered reduce-order  ( n f = 1 ) state estimator can reduce computational resources in by utilizing state estimators of smaller order instead of full-order state estimators to achieve the same goal in practical applications.

Remark 2
The reduced-order filter can save the communication resources because the order of the filter state is reduced.In detail, the state dimension of the full-order filter is equal to that of the original neural networks, while the state dimension of the reduced-order filter is less than that of the original neural networks.In terms of the  state dimension, the reduced-order filter saves communication resources.And in Fig. 6a, b, they are all eventtriggered filter.The event-triggered reduce-order ( n f = 1 ) state estimator can save communication resources because of the reduced order.Compared the event-triggered full-order filter ( n f = 21 ) in Fig. 6a with the event- triggered reduce-order ( n f = 1 ) state estimator in Fig. 6b, the reason of saving communication resources is same, i.e. the reduced order.However, for the filter without event-triggered strategy and the filter with event-triggered strategy, the filter that is not based on the event-triggered strategy receives is all y(t), while the filter based on the event-triggered strategy receives only the y(t) sifted by the event-triggered generator, thus realizing the saving of communication resources.Therefore, the reduce-order state estimator and the event-triggered reduce-order state estimator can reduce computational resources in by utilizing state estimators of smaller order instead of full-order state estimators.The method proposed in this paper has the following limitations: (1) In practical applications, it may be difficult to find reduced-order filters.(2) The eligible event-triggered strategy may miss the transmission of critical data, thus affecting the actual measurement and estimation.These are what we need to circumvent in our future research.
According to the above analysis, the proposed methods of the reduced-order H ∞ state estimators and the (reduced-order) H ∞ state estimators based on the event-triggered sampling strategy are availability.

Conclusions
An H ∞ state estimation has been studied for continuous-time neural networks with norm-bounded uncertainties via designing a reduced-order H ∞ state estimator and an event-triggered reduced-order H ∞ state estimator.For the considered neural networks with uncertainties, both a reduced-order H ∞ state estimator and an event-trig- gered reduced-order H ∞ state estimator have been designed, which ensure that the error dynamic is exponentially stable and has weighted H ∞ performance index by Lyapunov function method.Moreover, it has also given the devised methods of the reduced-order and event-triggered reduced-order H ∞ state estimator.compared with the full-order ( n f = 2 ) and reduce-order ( n f = 1 ) state estimators, the z f (t) of the reduce-order state estimator is closer to z(t), and the order is lower.The use of event-triggered sampling avoids the need to sample y(t) from time to time.Finally, by the example of well-known Tunnel Diode Circuit, it has shown that a lower order state estimator can be designed under the premise of maintaining the same weighted H ∞ performance index, and using the event-triggered sampling strategy can reduce the computational and time costs in communication and save resources.And in the future, our direction is focus on the reduce-order ( n f = 1 ) state estimators design for the neural networks with impulse and affine disturbance by applying an event-triggered strategy.

Figure 1 .
Figure 1.The Structure for State Estimator of the system (1).

Figure 2 .
Figure2.The structure for state estimator based on event-triggered sampling strategy (Eq.12).
The following Theorem is considered to get a smaller performance index.It's a suboptimal and better result in state estimation.